Operator Algebra On The Lie Map

The study of operator algebra theory began in the 1930s.With the fast development of the theory,now it has become a hot branch playing the role of an initiator in morden mathematics.It has unexpected relations and interinfiltrations with quantum mechanics,noncommutative geomtry,linear system and control theory, number theory as well as some other important branches of mathematics.In order to discuss the structure of operator algebras,in recent years,many scholars both here and abroad have focused on mappings of operator algebras.For example, derivations,biderivations,isomorphisms,element mappings,linear preserving mappings etc.They also have introduced some new concepts and new methods. For example,commuting mappings,functional identities etc.At present time these mappings have become important tools in studying operator algebras.Triangular algebra is a class of most important non-prime and non-selfadjoint algebra,such as upper triangular matrix algebra and nest algebra are this algebra.In this paper we mainly and detailedly discuss nonlinear Lie derivations,Lie derivable mappings at zero and generalized Lie derivable mappings at zero on some operator algebras. This paper contains four chapters.In chapter 1,some notations,definitions are introduced and some well-known results are given.In section 1.1,we introduce the definitions of nest algebras,Lie derivations,nonlinear Lie derivations,commuting mappings,Lie derivable mappings at zero,generalized *-Lie derivable mappings at zero and so on.In section 1.2,we give some well-known results.In chapter 2,we characterize nonlinear Lie derivations on matrix algebras.We prove that every nonlinear derivationφon matrix algebras is of the form A→4 AT—TA+A_ψ+h(A)I,whereψis an additive derivation,A_ψis the image of A underψapplied entrywise and h is a center-value nonlinear mapping and sending commutator to zero.In chapter 3,we first discuss Lie derivable mappings at zero on a nest algebra T∈τ(N),we prove that every Lie derivable mapping at zero on nest algebra is of the form A→AT—TA+λA+h(A),where Aλ∈(?),T∈τ(N) and h:τ(N)→(?)I is a linear mapping.And then we characterize generalized *-Lie derivable mappings at zero onβ(H),we prove every generalized Lie derivable mapping at zero onβ(H) has the form X→XT+T~*X,where T∈β(H) and T + T~* = cI,c∈(?). In chapter 4,we characterize nonlinear Lie derivations on nest algebras andβ(H).We prove that every nonlinear derivation on these algebras is the sum of an inner derivation and a center-valued nonlinear mapping sending commutators to zero.